free algebras via a functor on partial...
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![Page 1: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/1.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebras via a functoron partial algebras
Dion Coumans and Sam van Gool
Topology, Algebra and Categoriesin Logic (TACL)
26 – 30 July 2011Marseilles, France
1 / 16
![Page 2: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/2.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Logic via algebra
• Algebraic logic L, signature σ, variety VL of σ-algebras
• Studying the logic L! Studying finitely generated freeVL-algebras
LanguageFσ(x1, . . . , xm)
[·]a`L
LogicFVL(x1, . . . , xm)
2 / 16
![Page 3: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/3.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Logic via algebra
• Algebraic logic L, signature σ, variety VL of σ-algebras
• Studying the logic L! Studying finitely generated freeVL-algebras
LanguageFσ(x1, . . . , xm)
[·]a`L
LogicFVL(x1, . . . , xm)
2 / 16
![Page 4: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/4.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Logic via algebra
• Algebraic logic L, signature σ, variety VL of σ-algebras
• Studying the logic L! Studying finitely generated freeVL-algebras
LanguageFσ(x1, . . . , xm)
[·]a`L
LogicFVL(x1, . . . , xm)
2 / 16
![Page 5: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/5.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Logic via algebra
• Algebraic logic L, signature σ, variety VL of σ-algebras
• Studying the logic L! Studying finitely generated freeVL-algebras
LanguageFσ(x1, . . . , xm)
[·]a`L
LogicFVL(x1, . . . , xm)
2 / 16
![Page 6: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/6.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
• In many cases, a variety (VL)− of reducts iswell-understood and locally finite, e.g.:
• Modal algebras = Boolean algebras + ^,
• Heyting algebras = Distributive lattices +→,
• . . .
• Regard FVL(x1, . . . , xn) as colimit of a chain of finitealgebras in the reduced signature, and add the additionaloperation(s) step-by-step:
3 / 16
![Page 7: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/7.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
• In many cases, a variety (VL)− of reducts iswell-understood and locally finite, e.g.:
• Modal algebras = Boolean algebras + ^,
• Heyting algebras = Distributive lattices +→,
• . . .
• Regard FVL(x1, . . . , xn) as colimit of a chain of finitealgebras in the reduced signature, and add the additionaloperation(s) step-by-step:
3 / 16
![Page 8: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/8.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
• In many cases, a variety (VL)− of reducts iswell-understood and locally finite, e.g.:
• Modal algebras = Boolean algebras + ^,
• Heyting algebras = Distributive lattices +→,
• . . .
• Regard FVL(x1, . . . , xn) as colimit of a chain of finitealgebras in the reduced signature, and add the additionaloperation(s) step-by-step:
3 / 16
![Page 9: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/9.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
• In many cases, a variety (VL)− of reducts iswell-understood and locally finite, e.g.:
• Modal algebras = Boolean algebras + ^,
• Heyting algebras = Distributive lattices +→,
• . . .
• Regard FVL(x1, . . . , xn) as colimit of a chain of finitealgebras in the reduced signature, and add the additionaloperation(s) step-by-step:
3 / 16
![Page 10: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/10.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
• In many cases, a variety (VL)− of reducts iswell-understood and locally finite, e.g.:
• Modal algebras = Boolean algebras + ^,
• Heyting algebras = Distributive lattices +→,
• . . .
• Regard FVL(x1, . . . , xn) as colimit of a chain of finitealgebras in the reduced signature, and add the additionaloperation(s) step-by-step:
3 / 16
![Page 11: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/11.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
Language
· · ·Tn· · ·T1T0
[·]a`L
Logic
fff
· · ·Bn· · ·B1B0fff
• Tn: formulas in variables x1, . . . , xm of rank ≤ n inoperation f
• Bn: L-equivalence classes of formulas in Tn
FVL(x1, . . . , xm) = colimn≥0 Bn
4 / 16
![Page 12: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/12.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
Language
· · ·Tn· · ·T1
T0
[·]a`L
Logic
fff
· · ·Bn· · ·B1B0fff
• Tn: formulas in variables x1, . . . , xm of rank ≤ n inoperation f
• Bn: L-equivalence classes of formulas in Tn
FVL(x1, . . . , xm) = colimn≥0 Bn
4 / 16
![Page 13: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/13.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
Language
· · ·Tn· · ·T1
T0
[·]a`L
Logic
fff
· · ·Bn· · ·B1
B0
fff
• Tn: formulas in variables x1, . . . , xm of rank ≤ n inoperation f
• Bn: L-equivalence classes of formulas in Tn
FVL(x1, . . . , xm) = colimn≥0 Bn
4 / 16
![Page 14: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/14.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
Language
· · ·Tn· · ·
T1T0
[·]a`L
Logic
fff
· · ·Bn· · ·B1
B0
fff
• Tn: formulas in variables x1, . . . , xm of rank ≤ n inoperation f
• Bn: L-equivalence classes of formulas in Tn
FVL(x1, . . . , xm) = colimn≥0 Bn
4 / 16
![Page 15: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/15.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
Language
· · ·Tn· · ·
T1T0
[·]a`L
Logic
fff
· · ·Bn· · ·
B1B0
fff
• Tn: formulas in variables x1, . . . , xm of rank ≤ n inoperation f
• Bn: L-equivalence classes of formulas in Tn
FVL(x1, . . . , xm) = colimn≥0 Bn
4 / 16
![Page 16: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/16.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
Language
· · ·Tn
· · ·T1T0
[·]a`L
Logic
fff
· · ·Bn
· · ·B1B0
fff
• Tn: formulas in variables x1, . . . , xm of rank ≤ n inoperation f
• Bn: L-equivalence classes of formulas in Tn
FVL(x1, . . . , xm) = colimn≥0 Bn
4 / 16
![Page 17: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/17.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
Language
· · ·
Tn· · ·T1T0
[·]a`L
Logic
fff
· · ·
Bn· · ·B1B0
fff
• Tn: formulas in variables x1, . . . , xm of rank ≤ n inoperation f
• Bn: L-equivalence classes of formulas in Tn
FVL(x1, . . . , xm) = colimn≥0 Bn
4 / 16
![Page 18: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/18.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
Language· · ·Tn· · ·T1T0
[·]a`L
Logic
fff
· · ·Bn· · ·B1B0
fff
• Tn: formulas in variables x1, . . . , xm of rank ≤ n inoperation f
• Bn: L-equivalence classes of formulas in Tn
FVL(x1, . . . , xm) = colimn≥0 Bn
4 / 16
![Page 19: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/19.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
Language· · ·Tn· · ·T1T0
[·]a`L
Logic
f
ff
· · ·Bn· · ·B1B0
fff
• Tn: formulas in variables x1, . . . , xm of rank ≤ n inoperation f
• Bn: L-equivalence classes of formulas in Tn
FVL(x1, . . . , xm) = colimn≥0 Bn
4 / 16
![Page 20: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/20.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
Language· · ·Tn· · ·T1T0
[·]a`L
Logic
f
ff
· · ·Bn· · ·B1B0f
ff
• Tn: formulas in variables x1, . . . , xm of rank ≤ n inoperation f
• Bn: L-equivalence classes of formulas in Tn
FVL(x1, . . . , xm) = colimn≥0 Bn
4 / 16
![Page 21: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/21.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
Language· · ·Tn· · ·T1T0
[·]a`L
Logic
fff
· · ·Bn· · ·B1B0fff
• Tn: formulas in variables x1, . . . , xm of rank ≤ n inoperation f
• Bn: L-equivalence classes of formulas in Tn
FVL(x1, . . . , xm) = colimn≥0 Bn
4 / 16
![Page 22: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/22.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebra as colimit of a chain
Language· · ·Tn· · ·T1T0
[·]a`L
Logic
fff
· · ·Bn· · ·B1B0fff
• Tn: formulas in variables x1, . . . , xm of rank ≤ n inoperation f
• Bn: L-equivalence classes of formulas in Tn
FVL(x1, . . . , xm) = colimn≥0 Bn
4 / 16
![Page 23: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/23.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Research Question
· · ·Bn· · ·B1B0
Can Bn+1 be obtained from Bn by a uniform method?
• Yes, if the variety is defined by pure rank 1 equations[N. Bezhanishvili, Kurz]
• Yes, in some particular cases outside this class: S4 modalalgebras [Ghilardi], Heyting algebras [Ghilardi, N.Bezhanishvili & Gehrke].
• Not always, since logics can be undecidable.
• We give general sufficient conditions under which this ispossible (known cases follow as particular instances).
5 / 16
![Page 24: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/24.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Research Question
· · ·Bn· · ·B1B0
Can Bn+1 be obtained from Bn by a uniform method?
• Yes, if the variety is defined by pure rank 1 equations[N. Bezhanishvili, Kurz]
• Yes, in some particular cases outside this class: S4 modalalgebras [Ghilardi], Heyting algebras [Ghilardi, N.Bezhanishvili & Gehrke].
• Not always, since logics can be undecidable.
• We give general sufficient conditions under which this ispossible (known cases follow as particular instances).
5 / 16
![Page 25: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/25.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Research Question
· · ·Bn· · ·B1B0
Can Bn+1 be obtained from Bn by a uniform method?
• Yes, if the variety is defined by pure rank 1 equations[N. Bezhanishvili, Kurz]
• Yes, in some particular cases outside this class: S4 modalalgebras [Ghilardi], Heyting algebras [Ghilardi, N.Bezhanishvili & Gehrke].
• Not always, since logics can be undecidable.
• We give general sufficient conditions under which this ispossible (known cases follow as particular instances).
5 / 16
![Page 26: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/26.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Research Question
· · ·Bn· · ·B1B0
Can Bn+1 be obtained from Bn by a uniform method?
• Yes, if the variety is defined by pure rank 1 equations[N. Bezhanishvili, Kurz]
• Yes, in some particular cases outside this class: S4 modalalgebras [Ghilardi], Heyting algebras [Ghilardi, N.Bezhanishvili & Gehrke].
• Not always, since logics can be undecidable.
• We give general sufficient conditions under which this ispossible (known cases follow as particular instances).
5 / 16
![Page 27: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/27.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Research Question
· · ·Bn· · ·B1B0
Can Bn+1 be obtained from Bn by a uniform method?
• Yes, if the variety is defined by pure rank 1 equations[N. Bezhanishvili, Kurz]
• Yes, in some particular cases outside this class: S4 modalalgebras [Ghilardi], Heyting algebras [Ghilardi, N.Bezhanishvili & Gehrke].
• Not always, since logics can be undecidable.
• We give general sufficient conditions under which this ispossible (known cases follow as particular instances).
5 / 16
![Page 28: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/28.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Partial algebras
• In the chain, Bn+1 is a partial algebra, where the domainof the operation f is Bn.
• The variety V is contained in a category pV of partialalgebras for the variety V.
• A homomorphism h : A → B of partial algebras is afunction which preserves all total operations, andpreserves the partial operation f whenever defined.
• A homomorphism h : A → B is image-total if the image ofh is contained in the domain of fB .
6 / 16
![Page 29: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/29.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Partial algebras
• In the chain, Bn+1 is a partial algebra, where the domainof the operation f is Bn.
• The variety V is contained in a category pV of partialalgebras for the variety V.
• A homomorphism h : A → B of partial algebras is afunction which preserves all total operations, andpreserves the partial operation f whenever defined.
• A homomorphism h : A → B is image-total if the image ofh is contained in the domain of fB .
6 / 16
![Page 30: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/30.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Partial algebras
• In the chain, Bn+1 is a partial algebra, where the domainof the operation f is Bn.
• The variety V is contained in a category pV of partialalgebras for the variety V.
• A homomorphism h : A → B of partial algebras is afunction which preserves all total operations, andpreserves the partial operation f whenever defined.
• A homomorphism h : A → B is image-total if the image ofh is contained in the domain of fB .
6 / 16
![Page 31: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/31.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Partial algebras
• In the chain, Bn+1 is a partial algebra, where the domainof the operation f is Bn.
• The variety V is contained in a category pV of partialalgebras for the variety V.
• A homomorphism h : A → B of partial algebras is afunction which preserves all total operations, andpreserves the partial operation f whenever defined.
• A homomorphism h : A → B is image-total if the image ofh is contained in the domain of fB .
6 / 16
![Page 32: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/32.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorDefinition
DefinitionA functor F : pV→ pV is free image-total if there is acomponent-wise image-total natural transformation η : 1pV → Fsuch that, for all image-total h : A → B, there exists a uniqueh̄ : FA → B making the following diagram commute:
AηA - FA
B
h̄
?
h-
7 / 16
![Page 33: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/33.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorMain theorem
TheoremLet η : 1→ F be a free image-total functor and A0 ∈ pV. Let Aω
be the partial algebra-colimit of the image-total chain{ηFn(A0) : Fn(A0)→ Fn+1(A0)}n≥0.If Aω is in V, then Aω is the free total V-algebra over A0.
Proof.Category-theoretic arguments. �
Now, to apply this theorem:
• We construct a free image-total functor for any set ofquasi-equations,
• We give sufficient conditions under which Aω ∈ V.
8 / 16
![Page 34: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/34.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorMain theorem
TheoremLet η : 1→ F be a free image-total functor and A0 ∈ pV. Let Aω
be the partial algebra-colimit of the image-total chain{ηFn(A0) : Fn(A0)→ Fn+1(A0)}n≥0.If Aω is in V, then Aω is the free total V-algebra over A0.
Proof.Category-theoretic arguments. �
Now, to apply this theorem:
• We construct a free image-total functor for any set ofquasi-equations,
• We give sufficient conditions under which Aω ∈ V.
8 / 16
![Page 35: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/35.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorMain theorem
TheoremLet η : 1→ F be a free image-total functor and A0 ∈ pV. Let Aω
be the partial algebra-colimit of the image-total chain{ηFn(A0) : Fn(A0)→ Fn+1(A0)}n≥0.If Aω is in V, then Aω is the free total V-algebra over A0.
Proof.Category-theoretic arguments. �
Now, to apply this theorem:
• We construct a free image-total functor for any set ofquasi-equations,
• We give sufficient conditions under which Aω ∈ V.
8 / 16
![Page 36: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/36.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorMain theorem
TheoremLet η : 1→ F be a free image-total functor and A0 ∈ pV. Let Aω
be the partial algebra-colimit of the image-total chain{ηFn(A0) : Fn(A0)→ Fn+1(A0)}n≥0.If Aω is in V, then Aω is the free total V-algebra over A0.
Proof.Category-theoretic arguments. �
Now, to apply this theorem:
• We construct a free image-total functor for any set ofquasi-equations,
• We give sufficient conditions under which Aω ∈ V.
8 / 16
![Page 37: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/37.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorMain theorem
TheoremLet η : 1→ F be a free image-total functor and A0 ∈ pV. Let Aω
be the partial algebra-colimit of the image-total chain{ηFn(A0) : Fn(A0)→ Fn+1(A0)}n≥0.If Aω is in V, then Aω is the free total V-algebra over A0.
Proof.Category-theoretic arguments. �
Now, to apply this theorem:
• We construct a free image-total functor for any set ofquasi-equations,
• We give sufficient conditions under which Aω ∈ V.
8 / 16
![Page 38: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/38.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorConstruction
• Let E be a set of quasi-equations (of rank at most 1)axiomatizing the variety V.
• For A ∈ pV, define
FE(A) := [A + FV−(fA)]/θA
• V−: reduct of V to the signature of total operations,• fA : formal elements {fa : a ∈ A }, yielding partial operation
a 7→ fa for a ∈ A ,• θA : smallest pV-congruence on A + FV−(fA) containing〈fA a, fa〉, for all a ∈ dom(fA ).
• ηA is the composite
A � A + FV−(fA)� FE(A).
9 / 16
![Page 39: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/39.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorConstruction
• Let E be a set of quasi-equations (of rank at most 1)axiomatizing the variety V.
• For A ∈ pV, define
FE(A) := [A + FV−(fA)]/θA
• V−: reduct of V to the signature of total operations,• fA : formal elements {fa : a ∈ A }, yielding partial operation
a 7→ fa for a ∈ A ,• θA : smallest pV-congruence on A + FV−(fA) containing〈fA a, fa〉, for all a ∈ dom(fA ).
• ηA is the composite
A � A + FV−(fA)� FE(A).
9 / 16
![Page 40: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/40.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorConstruction
• Let E be a set of quasi-equations (of rank at most 1)axiomatizing the variety V.
• For A ∈ pV, define
FE(A) := [A + FV−(fA)]/θA
• V−: reduct of V to the signature of total operations,
• fA : formal elements {fa : a ∈ A }, yielding partial operationa 7→ fa for a ∈ A ,
• θA : smallest pV-congruence on A + FV−(fA) containing〈fA a, fa〉, for all a ∈ dom(fA ).
• ηA is the composite
A � A + FV−(fA)� FE(A).
9 / 16
![Page 41: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/41.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorConstruction
• Let E be a set of quasi-equations (of rank at most 1)axiomatizing the variety V.
• For A ∈ pV, define
FE(A) := [A + FV−(fA)]/θA
• V−: reduct of V to the signature of total operations,• fA : formal elements {fa : a ∈ A }, yielding partial operation
a 7→ fa for a ∈ A ,
• θA : smallest pV-congruence on A + FV−(fA) containing〈fA a, fa〉, for all a ∈ dom(fA ).
• ηA is the composite
A � A + FV−(fA)� FE(A).
9 / 16
![Page 42: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/42.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorConstruction
• Let E be a set of quasi-equations (of rank at most 1)axiomatizing the variety V.
• For A ∈ pV, define
FE(A) := [A + FV−(fA)]/θA
• V−: reduct of V to the signature of total operations,• fA : formal elements {fa : a ∈ A }, yielding partial operation
a 7→ fa for a ∈ A ,• θA : smallest pV-congruence on A + FV−(fA) containing〈fA a, fa〉, for all a ∈ dom(fA ).
• ηA is the composite
A � A + FV−(fA)� FE(A).
9 / 16
![Page 43: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/43.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorConstruction
• Let E be a set of quasi-equations (of rank at most 1)axiomatizing the variety V.
• For A ∈ pV, define
FE(A) := [A + FV−(fA)]/θA
• V−: reduct of V to the signature of total operations,• fA : formal elements {fa : a ∈ A }, yielding partial operation
a 7→ fa for a ∈ A ,• θA : smallest pV-congruence on A + FV−(fA) containing〈fA a, fa〉, for all a ∈ dom(fA ).
• ηA is the composite
A � A + FV−(fA)� FE(A).
9 / 16
![Page 44: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/44.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorLemma
LemmaFE is a free image-total functor with universal arrow η.Furthermore, if A0 ∈ pV is such that each componentηFnE(A0) : Fn
E(A0)→ Fn+1
E(A0) is an embedding, then Aω ∈ V.
Proof.Uses universal algebra for partial algebras. �
CorollaryIf A0 ∈ pV is such that each componentηFnE(A0) : Fn
E(A0)→ Fn+1
E(A0) is an embedding, then Aω is the
free total V-algebra over A0.
10 / 16
![Page 45: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/45.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorLemma
LemmaFE is a free image-total functor with universal arrow η.Furthermore, if A0 ∈ pV is such that each componentηFnE(A0) : Fn
E(A0)→ Fn+1
E(A0) is an embedding, then Aω ∈ V.
Proof.Uses universal algebra for partial algebras. �
CorollaryIf A0 ∈ pV is such that each componentηFnE(A0) : Fn
E(A0)→ Fn+1
E(A0) is an embedding, then Aω is the
free total V-algebra over A0.
10 / 16
![Page 46: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/46.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free image-total functorLemma
LemmaFE is a free image-total functor with universal arrow η.Furthermore, if A0 ∈ pV is such that each componentηFnE(A0) : Fn
E(A0)→ Fn+1
E(A0) is an embedding, then Aω ∈ V.
Proof.Uses universal algebra for partial algebras. �
CorollaryIf A0 ∈ pV is such that each componentηFnE(A0) : Fn
E(A0)→ Fn+1
E(A0) is an embedding, then Aω is the
free total V-algebra over A0.
10 / 16
![Page 47: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/47.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The variety KB
• Signature = ⊥,>,∨,∧,¬,^
• Axioms = Boolean algebras +
^⊥ = ⊥
^(x ∨ y) = ^x ∨ ^y
x ≤ ¬^y → y ≤ ¬^x.
• (Finite) Duality theory:
• KB algebras↔ Sets with a symmetric relation• Partial KB algebras↔ Sets with an equivalence relation ∼
and a quasi-symmetric relation R satisfying R ◦ ∼⊆ R
11 / 16
![Page 48: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/48.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The variety KB
• Signature = ⊥,>,∨,∧,¬,^
• Axioms = Boolean algebras +
^⊥ = ⊥
^(x ∨ y) = ^x ∨ ^y
x ≤ ¬^y → y ≤ ¬^x.
• (Finite) Duality theory:
• KB algebras↔ Sets with a symmetric relation• Partial KB algebras↔ Sets with an equivalence relation ∼
and a quasi-symmetric relation R satisfying R ◦ ∼⊆ R
11 / 16
![Page 49: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/49.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The variety KB
• Signature = ⊥,>,∨,∧,¬,^
• Axioms = Boolean algebras +
^⊥ = ⊥
^(x ∨ y) = ^x ∨ ^y
x ≤ ¬^y → y ≤ ¬^x.
• (Finite) Duality theory:
• KB algebras↔ Sets with a symmetric relation• Partial KB algebras↔ Sets with an equivalence relation ∼
and a quasi-symmetric relation R satisfying R ◦ ∼⊆ R
11 / 16
![Page 50: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/50.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The variety KB
• Signature = ⊥,>,∨,∧,¬,^
• Axioms = Boolean algebras +
^⊥ = ⊥
^(x ∨ y) = ^x ∨ ^y
x ≤ ¬^y → y ≤ ¬^x.
• (Finite) Duality theory:• KB algebras↔ Sets with a symmetric relation
• Partial KB algebras↔ Sets with an equivalence relation ∼and a quasi-symmetric relation R satisfying R ◦ ∼⊆ R
11 / 16
![Page 51: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/51.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The variety KB
• Signature = ⊥,>,∨,∧,¬,^
• Axioms = Boolean algebras +
^⊥ = ⊥
^(x ∨ y) = ^x ∨ ^y
x ≤ ¬^y → y ≤ ¬^x.
• (Finite) Duality theory:• KB algebras↔ Sets with a symmetric relation• Partial KB algebras↔ Sets with an equivalence relation ∼
and a quasi-symmetric relation R satisfying R ◦ ∼⊆ R
11 / 16
![Page 52: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/52.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The functor FKB
• By definition, for a partial KB algebra A ,
FKB(A) = [A + FBA(_A)]/θA .
GKB(X)
A
Duality
X
FBA(_A)+
× P(X)
A FBA(_A)+/θA
• Using correspondence theory, one can explicitly calculatea first-order definition of the points in GKB(X ,R ,∼)
• These points are normal forms for KB.
12 / 16
![Page 53: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/53.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The functor FKB
• By definition, for a partial KB algebra A ,
FKB(A) = [A + FBA(_A)]/θA .
GKB(X)
A
Duality
X
FBA(_A)+
× P(X)
A FBA(_A)+/θA
• Using correspondence theory, one can explicitly calculatea first-order definition of the points in GKB(X ,R ,∼)
• These points are normal forms for KB.
12 / 16
![Page 54: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/54.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The functor FKB
• By definition, for a partial KB algebra A ,
FKB(A) = [A + FBA(_A)]/θA .
GKB(X)
A
Duality
X
FBA(_A)+
× P(X)
A FBA(_A)+/θA
• Using correspondence theory, one can explicitly calculatea first-order definition of the points in GKB(X ,R ,∼)
• These points are normal forms for KB.
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Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The functor FKB
• By definition, for a partial KB algebra A ,
FKB(A) = [A + FBA(_A)]/θA .
GKB(X)
A
Duality
X
FBA(_A)+
× P(X)
A FBA(_A)+/θA
• Using correspondence theory, one can explicitly calculatea first-order definition of the points in GKB(X ,R ,∼)
• These points are normal forms for KB.
12 / 16
![Page 56: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/56.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The functor FKB
• By definition, for a partial KB algebra A ,
FKB(A) = [A + FBA(_A)]/θA .
GKB(X)
A
Duality
X
FBA(_A)+
× P(X)
A FBA(_A)+/θA
• Using correspondence theory, one can explicitly calculatea first-order definition of the points in GKB(X ,R ,∼)
• These points are normal forms for KB.
12 / 16
![Page 57: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/57.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The functor FKB
• By definition, for a partial KB algebra A ,
FKB(A) = [A + FBA(_A)]/θA .
GKB(X)
A
Duality
X
FBA(_A)+
× P(X)
A FBA(_A)+/θA
• Using correspondence theory, one can explicitly calculatea first-order definition of the points in GKB(X ,R ,∼)
• These points are normal forms for KB.12 / 16
![Page 58: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/58.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The chain for KBFirst steps
X0
p ¬p
GKB(X0)
p ∧ ^p ∧ ^¬p
p ∧ ^p ∧ ¬^¬p
p ∧ ¬^p ∧ ^¬p
p ∧ ¬^p ∧ ¬^¬p
¬p ∧ ^p ∧ ^¬p
¬p ∧ ^p ∧ ¬^¬p
¬p ∧ ¬^p ∧ ^¬p
¬p ∧ ¬^p ∧ ¬^¬p
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![Page 59: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/59.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The chain for KBFirst steps
X0
p ¬p
GKB(X0)
p ∧ ^p ∧ ^¬p
p ∧ ^p ∧ ¬^¬p
p ∧ ¬^p ∧ ^¬p
p ∧ ¬^p ∧ ¬^¬p
¬p ∧ ^p ∧ ^¬p
¬p ∧ ^p ∧ ¬^¬p
¬p ∧ ¬^p ∧ ^¬p
¬p ∧ ¬^p ∧ ¬^¬p
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![Page 60: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/60.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The chain for KB(part of) G2
KB(X0)
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Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The chain for S4First steps
X0
GS4(X0)
G2S4(X0)
(...)
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Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The chain for S4First steps
X0
GS4(X0)
G2S4(X0)
(...)
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![Page 63: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/63.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The chain for S4First steps
X0
GS4(X0)
G2S4(X0)
(...)
15 / 16
![Page 64: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/64.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
The chain for S4First steps
X0
GS4(X0)
G2S4(X0)
(...)
15 / 16
![Page 65: Free algebras via a functor on partial algebraspageperso.lif.univ-mrs.fr/~luigi.santocanale/tacl2011/slides/76.pdf · Free alg’s via functor on partial alg’s Dion Coumans and](https://reader030.vdocument.in/reader030/viewer/2022040209/5e440e4be3bfda29ed597a16/html5/thumbnails/65.jpg)
Free alg’s viafunctor on
partial alg’s
Dion Coumansand
Sam van Gool
Free algebrastep-by-step
Freeimage-totalfunctor
Application toKB
Free algebras via a functoron partial algebras
Dion Coumans and Sam van Gool
Topology, Algebra and Categoriesin Logic (TACL)
26 – 30 July 2011Marseilles, France
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